I'm led to believe that this is mostly apocryphal, and not something that actually happened at a meaningful scale.
That specific story and No Irish Need Apply, yes. (Edit: Yes, it is apocryphal.)
Irish having been actual slaves (in the Thirteen Colonies) and being non-White, complete bullshit.
Them being viewed as a shitty low class criminal scumbag minority coming from a civilization (Catholic) incompatible with American principles, absolutely true. Like someone else mentioned, they were the Mexicans of the mid 1800s, then Italians were the Mexicans of the late 1800s/early 1900s. There is nothing new under the Sun.
Where I differ from both Right and Left is I see both the Irish and Italians as worse than most people see them today, and the Mexicans as better. These immigrants waves cause lots of genuine problems but the state always assimilates them and is better off for it, GIVEN that they have the shit kicked out of them to make them learn to behave. Italians might have actually been worse than Mexicans overall. There is no Know Nothings, Ku Klux Klan or any other kind of militant nativist
movement to bust out clubs and rope as the stick part of carrot and stick. But the need for such a thing is also greatly reduced in the countryside.
I also do not believe this applies to (most) Muslims, whose religion, in its present form, is too alien and aggressive. Muslims always act, at group level at least, deceptively and maliciously. I don’t believe Eastern European Jews (in contrast to the early Antebellum mass migration of German Jews) were worth letting in either, and for similar reasons.
Indians, it sounds like, were sold to us as being the new Germans (model minority) and like the Chinese, but are starting to reveal themselves for a bunch of barbarians. Most Indians used to be things like Sikhs and Gujarati Pindars, and India is a Europes worth of diversity, so we’re starting to get the absolute shit tier of its people.